Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New 53. and Applications to Risk Measures

Transcription

1 Law of Large Number and Central Limit Theorem under Uncertainty, the Related New Itô s Calculus and Applications to Risk Measures Shige Peng Shandong University, Jinan, China Presented at 1st PRIMA Congress, July 06-10, 2009 UNSW, Sydney, Australia

2 A Chinese Understanding of Beijing Opera A popular point of view of Beijing opera by Chinese artists,

3 A Chinese Understanding of Beijing Opera A popular point of view of Beijing opera by Chinese artists, Translation (with certain degree of uncertainty) The universe is a big opera stage, An opera stage is a small universe

4 A Chinese Understanding of Beijing Opera A popular point of view of Beijing opera by Chinese artists, Translation (with certain degree of uncertainty) The universe is a big opera stage, An opera stage is a small universe Real world Modeling world

5 Scientists point of view of the real world after Newton: People strongly believe that our universe is deterministic: it can be precisely calculated by (ordinary) differential equations.

6 Scientists point of view of the real world after Newton: People strongly believe that our universe is deterministic: it can be precisely calculated by (ordinary) differential equations. After longtime explore our scientists begin to realize that our universe is full of uncertainty

7 A fundamental tool to understand and to treat uncertainty: Probability Theory Pascal & Fermat, Huygens, de Moivre, Laplace, Ω: a set of events: only one event ω Ω happens but we don t know which ω (at least before it happens).

8 A fundamental tool to understand and to treat uncertainty: Probability Theory Pascal & Fermat, Huygens, de Moivre, Laplace, Ω: a set of events: only one event ω Ω happens but we don t know which ω (at least before it happens). The probability of A Ω: P(A) number times A happens total number of trials

9 A fundamental revolution of probability theory (1933 Kolmogorov s Foundation of Probability Theory (Ω, F, P)

10 A fundamental revolution of probability theory (1933 Kolmogorov s Foundation of Probability Theory (Ω, F, P) The basic rule of probability theory P( P(Ω) = 1, P( ) = 0, P(A + B) = P(A) + P(B) i=1 A i )= P(A i ) i=1

11 A fundamental revolution of probability theory (1933 Kolmogorov s Foundation of Probability Theory (Ω, F, P) The basic rule of probability theory P( P(Ω) = 1, P( ) = 0, P(A + B) = P(A) + P(B) i=1 A i )= P(A i ) i=1 The basic idea of Kolmogorov: Using (possibly infinite dimensional) Lebesgue integral to calculate the expectation of a random variable X (ω) E [X ]= Ω X (ω)dp(ω)

12 vnm Expected Utility Theory: The foundation of modern economic theory E [u(x )] E [u(y )] The utility of X is bigger than that of Y

13 Probability Theory Markov Processes Itô s calculus: and pathwise stochastic analysis Statistics: life science and medical industry; insurance, politics; Stochastic controls; Statistic Physics; Economics, Finance; Civil engineering; Communications, internet; Forecasting: Whether, pollution,

14 Probability measure P( ) vs mathematical expectation E [ ]: who is more fundamental? We can also first define a linear functional E : H R, s.t. E [αx + c] = αe [X ]+c, E [X ] E [Y ], if X Y E [X i ] 0, if X i 0.

15 Probability measure P( ) vs mathematical expectation E [ ]: who is more fundamental? We can also first define a linear functional E : H R, s.t. E [αx + c] = αe [X ]+c, E [X ] E [Y ], if X Y E [X i ] 0, if X i 0. By Daniell-Stone Theorem There is a unique probability measure (Ω, F, P) such that P(A) =E [1 A ] for each A F

16 Probability measure P( ) vs mathematical expectation E [ ]: who is more fundamental? We can also first define a linear functional E : H R, s.t. E [αx + c] = αe [X ]+c, E [X ] E [Y ], if X Y E [X i ] 0, if X i 0. By Daniell-Stone Theorem There is a unique probability measure (Ω, F, P) such that P(A) =E [1 A ] for each A F For nonlinear expectation No more such equivalence!!

17 Why nonlinear expectation? M. Allais Paradox Ellesberg Paradox in economic theory The linearity of expectation E cannot be a linear operator [ Artzner-Delbean-Eden-Heath] Coherent risk measure

18 A well-known puzzle: why normal distributions are so widely used? If someone faces a random variable X with uncertain distribution, he will first try to use normal distribution N (µ, σ 2 ).

19 The density of anormal distribution p(x) = 1 2π exp x2 2

20 Explanation for this phenomenon is the well-known central limit theorem:

21 Explanation for this phenomenon is the well-known central limit theorem: Theorem (Central Limit Theorem (CLT)) {X i } i=1 is assumed to be i.i.d. with µ = E [X 1] and σ 2 = E [(X 1 µ) 2 ]. Then for each bounded and continuous function ϕ C (R), we have lim E[ϕ( 1 n i n i µ))] = E [ϕ(x )], X N(0, σ i=1(x 2 ).

22 Explanation for this phenomenon is the well-known central limit theorem: Theorem (Central Limit Theorem (CLT)) {X i } i=1 is assumed to be i.i.d. with µ = E [X 1] and σ 2 = E [(X 1 µ) 2 ]. Then for each bounded and continuous function ϕ C (R), we have lim E[ϕ( 1 n i n i µ))] = E [ϕ(x )], X N(0, σ i=1(x 2 ). The beauty and power of this result comes from: the above sum tends to N (0, σ 2 ) regardless the original distribution of X i, provided that X i X 1, for all i = 2, 3, and that X 1, X 2, are mutually independent.

23 History of CLT Abraham de Moivre 1733, Pierre-Simon Laplace, 1812: Théorie Analytique des Probabilité Aleksandr Lyapunov, 1901 Cauchy s, Bessel s and Poisson s contributions, von Mises, Polya, Lindeberg, Lévy, Cramer

24 dirty work through contaminated data? But in, real world in finance, as well as in many other human science situations, it is not true that the above {X i } i=1 is really i.i.d. Many academic people think that people in finance just widely and deeply abuse this beautiful mathematical result to do dirty work through contaminated data

25 A new CLT under Distribution Uncertainty 1. We do not know the distribution of X i ; 2. We don t assume X i X j, they may have difference unknown distributions. We only assume that the distribution of X i, i = 1, 2, are within some subset of distribution functions L(X i ) {F θ (x) : θ Θ}.

26 In the situation of distributional uncertainty and/or probabilistic uncertainty (or model uncertainty) the problem of decision become more complicated. A well-accepted method is the following robust calculation: sup E θ [ϕ(ξ)], inf E θ[ϕ(η)] θ Θ θ Θ where E θ represent the expectation of a possible probability in our uncertainty model.

27 But this turns out to be a new and basic mathematical tool: Ê[X ]=sup E θ [X ] θ Θ The operator Ê has the properties: a) X Y then Ê[X ] Ê[Y ] b) Ê[c] = c c) Ê[X + Y ] Ê[X ]+Ê[Y ] d) Ê[λX ]=λê[x ], λ 0.

28 Sublinear Expectation Definition A nonlinear expectation: a functional Ê : H R

29 Sublinear Expectation Definition A nonlinear expectation: a functional Ê : H R (a) Monotonicity: if X Y then Ê[X ] Ê[Y ].

30 Sublinear Expectation Definition A nonlinear expectation: a functional Ê : H R (a) Monotonicity: if X Y then Ê[X ] Ê[Y ]. (b) Constant preserving: Ê[c] =c. A sublinear expectation:

31 Sublinear Expectation Definition A nonlinear expectation: a functional Ê : H R (a) Monotonicity: if X Y then Ê[X ] Ê[Y ]. (b) Constant preserving: Ê[c] =c. A sublinear expectation: (c) Sub-additivity (or self dominated property): Ê[X ] Ê[Y ] Ê[X Y ].

32 Sublinear Expectation Definition A nonlinear expectation: a functional Ê : H R (a) Monotonicity: if X Y then Ê[X ] Ê[Y ]. (b) Constant preserving: Ê[c] =c. A sublinear expectation: (c) Sub-additivity (or self dominated property): Ê[X ] Ê[Y ] Ê[X Y ]. (d) Positive homogeneity: Ê[λX ]=λê[x], λ 0.

33 Coherent Risk Measures and Sunlinear Expectations H is the set of bounded and measurable random variables on (Ω, F). ρ(x ) := Ê[ X ]

34 Robust representation of sublinear expectations Huber Robust Statistics (1981). Artzner, Delbean, Eber & Heath (1999) Föllmer & Schied (2004) (For interest rate uncertainty, see Barrieu & El Karoui (2005)).

35 Robust representation of sublinear expectations Theorem Huber Robust Statistics (1981). Artzner, Delbean, Eber & Heath (1999) Föllmer & Schied (2004) Ê[ ] is a sublinear expectation on H if and only if there exists a subset P M 1,f (the collection of all finitely additive probability measures) such that Ê[X ]=sup E P [X ], X H. P P (For interest rate uncertainty, see Barrieu & El Karoui (2005)).

36 Meaning of the robust representation: Statistic model uncertainty Ê[X ] = sup P P E P [X ], X H. The size of the subset P represents the degree of model uncertainty: The stronger the Ê the more the uncertainty Ê 1 [X ] Ê 2 [X ], X H P 1 P 2

37 The distribution of a random vector in (Ω, H, Ê)

38 The distribution of a random vector in (Ω, H, Ê) Definition (Distribution of X ) Given X = (X 1,, X n ) H n. We define: ˆF X [ϕ] := Ê[ϕ(X )] : ϕ C b (R n ) R. We call ˆF X [ ] the distribution of X under Ê.

39 The distribution of a random vector in (Ω, H, Ê) Definition (Distribution of X ) Given X = (X 1,, X n ) H n. We define: ˆF X [ϕ] := Ê[ϕ(X )] : ϕ C b (R n ) R. We call ˆF X [ ] the distribution of X under Ê. Sublinear Distribution F X [ ] forms a sublinear expectation on (R n, C b (R n )). Briefly: F X [ϕ] =sup ϕ(x)f θ (dy). θ Θ R n

40 Definition Definition X,Y are said to be identically distributed, (X Y,orX is a copy of Y ), if they have same distributions: Ê[ϕ(X )] = Ê[ϕ(Y )], ϕ C b (R n ).

41 Definition Definition X,Y are said to be identically distributed, (X Y,orX is a copy of Y ), if they have same distributions: Ê[ϕ(X )] = Ê[ϕ(Y )], ϕ C b (R n ). The meaning of X Y : X and Y has the same subset of uncertain distributions.

42 Independence under Ê Definition Y (ω) is said to be independent from X (ω) if we have: Ê[ϕ(X, Y )] = Ê[Ê[ϕ(x, Y )] x=x ], ϕ( ).

43 Independence under Ê Definition Y (ω) is said to be independent from X (ω) if we have: Ê[ϕ(X, Y )] = Ê[Ê[ϕ(x, Y )] x=x ], ϕ( ). Fact Meaning: the realization of X does not change (improve) the distribution uncertainty of Y.

44 The notion of independence: it can be subjective Example (An extreme example) In reality, Y = X but we are in the very beginning and we know noting about the relation of X and Y, the only information we know is X, Y Θ. The robust expectation of ϕ(x, Y ) is: Ê[ϕ(X, Y )] = sup x,y Θ ϕ(x, y). Y is independent to X, X is also independent to Y.

45 The notion of independence Fact Y is independent to X X is independent to Y Example DOES NOT IMPLIES σ 2 := Ê[Y 2 ] > σ 2 := Ê[ Y 2 ] > 0, Ê[X ] = Ê[ X ]=0. Then

46 The notion of independence Fact Y is independent to X X is independent to Y Example DOES NOT IMPLIES σ 2 := Ê[Y 2 ] > σ 2 := Ê[ Y 2 ] > 0, Ê[X ] = Ê[ X ]=0. Then If Y is independent to X : Ê[XY 2 ]=Ê[Ê[xY 2 ] x=x ]=Ê[X + σ 2 X σ 2 ] = Ê[X + ](σ 2 σ 2 ) > 0.

47 The notion of independence Fact Y is independent to X X is independent to Y Example DOES NOT IMPLIES σ 2 := Ê[Y 2 ] > σ 2 := Ê[ Y 2 ] > 0, Ê[X ] = Ê[ X ]=0. Then If Y is independent to X : Ê[XY 2 ]=Ê[Ê[xY 2 ] x=x ]=Ê[X + σ 2 X σ 2 ] But if X is independent to Y : = Ê[X + ](σ 2 σ 2 ) > 0. Ê[XY 2 ]=Ê[Ê[X ]Y 2 ]=0.

48 Central Limit Theorem (CLT) Let {X i } i=1 be a i.i.d. in a sublinear expectation space (Ω, H, Ê) in the following sense: (i) identically distributed: Ê[ϕ(X i )] = Ê[ϕ(X 1 )], i = 1, 2, (ii) independent: for each i, X i+1 is independent to (X 1, X 2,, X i ) under Ê. We also assume that Ê[X 1 ]= Ê[ X 1 ]. We denote σ 2 = Ê[X 2 1 ], σ 2 = Ê[ X 2 1 ] Then for each convex function ϕ we have Ê[ϕ( S n n )] 1 2πσ 2 ϕ(x) exp( x 2 2σ 2 )dx and for each concave function ψ we have S n 1 x 2

49 Theorem ( CLT under distribution uncertainty) Let {X i } i=1 be a i.i.d. in a sublinear expectation space (Ω, H, Ê) in the following sense: (i) identically distributed: Ê[ϕ(X i )] = Ê[ϕ(X 1 )], i = 1, 2, (ii) independent: for each i, X i+1 is independent to (X 1, X 2,, X i ) under Ê. We also assume that Ê[X 1 ]= Ê[ X 1 ]. Then for each convex function ϕ we have Ê[ϕ( S n n )] Ê[ϕ(X )], X N(0, [σ 2, σ 2 ])

50 Normal distribution under Sublinear expectation An fundamentally important sublinear distribution Definition A random variable X in (Ω, H, Ê) is called normally distributed if ax + b X a 2 + b 2 X, a, b 0. where X is an independent copy of X.

51 Normal distribution under Sublinear expectation An fundamentally important sublinear distribution Definition A random variable X in (Ω, H, Ê) is called normally distributed if ax + b X a 2 + b 2 X, a, b 0. where X is an independent copy of X. We have Ê[X ]=Ê[ X]=0.

52 Normal distribution under Sublinear expectation An fundamentally important sublinear distribution Definition A random variable X in (Ω, H, Ê) is called normally distributed if ax + b X a 2 + b 2 X, a, b 0. where X is an independent copy of X. We have Ê[X ]=Ê[ X ]=0. We denote X N(0, [σ 2, σ 2 ]), where σ 2 := Ê[X 2 ], σ 2 := Ê[ X 2 ].

53 G-normal distribution: a under sublinear expectation E[ ] (1) For each convex ϕ, we have Ê[ϕ(X )] = 1 2πσ 2 ϕ(y) exp( y 2 2σ 2 )dy

54 G-normal distribution: a under sublinear expectation E[ ] (1) For each convex ϕ, we have Ê[ϕ(X )] = 1 2πσ 2 ϕ(y) exp( y 2 2σ 2 )dy (2) For each concave ϕ, we have, Ê[ϕ(X )] = 1 2πσ 2 ϕ(y) exp( y 2 2σ 2 )dy

55 Fact If σ 2 = σ 2, then N (0; [σ 2, σ 2 ])=N(0, σ 2 ). Fact The larger to [σ 2, σ 2 ] the stronger the uncertainty. Fact But the uncertainty subset of N (0; [σ 2, σ 2 ]) is not just consisting of N (0; σ), σ [σ 2, σ 2 ]!!

56 Theorem X N (0, [σ 2, σ 2 ]) in (Ω, H, Ê) iff for each ϕ C b (R) the function u(t, x) := Ê[ϕ(x + tx )], x R, t 0 is the solution of the PDE u t = G (u xx ), t > 0, x R u t=0 = ϕ, where G (a) = 1 2 (σ2 a + σ 2 a )(= Ê[ a 2 X 2 ]). G-normal distribution.

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58 Law of Large Numbers (LLN), Central Limit Theorem (CLT) Striking consequence of LLN & CLT Accumulated independent and identically distributed random variables tends to a normal distributed random variable, whatever the original distribution.

59 Law of Large Numbers (LLN), Central Limit Theorem (CLT) Striking consequence of LLN & CLT Accumulated independent and identically distributed random variables tends to a normal distributed random variable, whatever the original distribution. LLN under Choquet capacities: Marinacci, M. Limit laws for non-additive probabilities and their frequentist interpretation, Journal of Economic Theory 84, Nothing found for nonlinear CLT.

60 Definition A sequence of random variables {η i } i=1 under Ê if the limit in H is said to converge in law lim Ê[ϕ(η i )], i for each ϕ C b (R).

61 Central Limit Theorem under Sublinear Expectation Theorem Let {X i } i=1 in (Ω, H, Ê) be identically distributed: X i X 1, and each X i+1 is independent to (X 1,, X i ). We assume furthermore that Ê[ X 1 2+α ] < and Ê[X 1 ]=Ê[ X 1 ]=0. S n := X X n. Then S n / n converges in law to N (0; [σ 2, σ 2 ]): where lim Ê[ϕ( S n )]=Ê[ϕ(X)], ϕ C b (R), n n where X N (0, [σ 2, σ 2 ]), σ 2 = Ê[X 2 1 ], σ 2 = Ê[ X 2 1 ].

62 Cases with mean-uncertainty What happens if Ê[X 1 ] > Ê[ X 1 ]? Definition A random variable Y in (Ω, H, Ê) is N ([µ, µ] {0})-distributed (Y N([µ, µ] {0})) if ay + bȳ (a + b)y, a, b 0. where Ȳ is an independent copy of Y, where µ := Ê[Y ] > µ := Ê[ Y ]

63 Cases with mean-uncertainty What happens if Ê[X 1 ] > Ê[ X 1 ]? Definition A random variable Y in (Ω, H, Ê) is N ([µ, µ] {0})-distributed (Y N([µ, µ] {0})) if ay + bȳ (a + b)y, a, b 0. where Ȳ is an independent copy of Y, where µ := Ê[Y ] > µ := Ê[ Y ] We can prove that Ê[ϕ(Y )] = sup y [µ,µ] ϕ(y).

64 Case with mean-uncertainty E[ ] Definition A pair of random variables (X, Y ) in (Ω, H, Ê) is N ([µ, µ] [σ 2, σ 2 ])-distributed ((X, Y ) N([µ, µ] [σ 2, σ 2 ])) if (ax + b X, a 2 Y + b 2 Ȳ ) ( a 2 + b 2 X, (a 2 + b 2 )Y ), a, b 0. where ( X, Ȳ ) is an independent copy of (X, Y ), µ := Ê[Y ], µ := Ê[ Y ] σ 2 := Ê[X 2 ], σ 2 := Ê[ X ], (Ê[X ] = Ê[ X ]=0).

65 Theorem (X, Y ) N ([µ, µ], [σ 2, σ 2 ]) in (Ω, H, Ê) iff for each ϕ C b (R) the function u(t, x, y) := Ê[ϕ(x + tx, y + ty )], x R, t 0 is the solution of the PDE u t = G (u y, u xx ), t > 0, x R u t=0 = ϕ, where G (p, a) := Ê[ a 2 X 2 + py ].

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67 Central Limit Theorem under Sublinear Expectation Theorem Let {X i + Y i } i=1 be an independent and identically distributed sequence. We assume furthermore that Ê[ X 1 2+α ]+Ê[ Y 1 1+α ] < and Ê[X 1 ]=Ê[ X 1 ]=0. S X n := X X n,s Y n := Y Y n. Then S n / n converges in law to N (0; [σ 2, σ 2 ]): lim Ê[ϕ( S n X + S n Y n n n )]=Ê[ϕ(X + Y )], ϕ C b (R), where (X, Y ) is N ([µ, µ], [σ 2, σ 2 ])-distributed.

68 Remarks. 1. Like classical case, these new LLN and CLT plays fundamental roles for the case with probability and distribution uncertainty;

69 Remarks. 1. Like classical case, these new LLN and CLT plays fundamental roles for the case with probability and distribution uncertainty; 2. According to our LLN and CLT, the uncertainty accumulates, in general, there are infinitely many probability measures. The best way to calculate Ê[X ] is to use our theorem, instead of calculating Ê[X ]=sup θ Θ E θ [X ];

70 Remarks. 1. Like classical case, these new LLN and CLT plays fundamental roles for the case with probability and distribution uncertainty; 2. According to our LLN and CLT, the uncertainty accumulates, in general, there are infinitely many probability measures. The best way to calculate Ê[X ] is to use our theorem, instead of calculating Ê[X ]=sup θ Θ E θ [X ]; 3. Many statistic methods must be revisited to clarify their meaning of uncertainty: it s very risky to treat this new notion in a classical way. This will cause serious confusions.

71 Remarks. 1. Like classical case, these new LLN and CLT plays fundamental roles for the case with probability and distribution uncertainty; 2. According to our LLN and CLT, the uncertainty accumulates, in general, there are infinitely many probability measures. The best way to calculate Ê[X ] is to use our theorem, instead of calculating Ê[X ]=sup θ Θ E θ [X ]; 3. Many statistic methods must be revisited to clarify their meaning of uncertainty: it s very risky to treat this new notion in a classical way. This will cause serious confusions. 4. The third revolution: Deterministic point of view = Probabilistic point of view = uncertainty of probabilities.

72 G Brownian Motion Definition Under (Ω, F, Ê), a process B t (ω) =ω t, t 0, is called a G Brownian motion if: (i) B t+s B s is N (0, [σ 2 t, σ 2 t]) distributed s, t 0

73 G Brownian Motion Definition Under (Ω, F, Ê), a process B t (ω) =ω t, t 0, is called a G Brownian motion if: (i) B t+s B s is N (0, [σ 2 t, σ 2 t]) distributed s, t 0 (ii) For each t 1 t n, B tn B tn 1 (B t1,, B tn 1 ). is independent to

74 G Brownian Motion Definition Under (Ω, F, Ê), a process B t (ω) =ω t, t 0, is called a G Brownian motion if: (i) B t+s B s is N (0, [σ 2 t, σ 2 t]) distributed s, t 0 (ii) For each t 1 t n, B tn B tn 1 (B t1,, B tn 1 ). For simplification, we set σ 2 = 1. is independent to

75 Theorem If, under some (Ω, F, Ê), a stochastic process B t (ω), t 0 satisfies

76 Theorem If, under some (Ω, F, Ê), a stochastic process B t (ω), t 0 satisfies For each t 1 t n, B tn B tn 1 is independent to (B t1,, B tn 1 ).

77 Theorem If, under some (Ω, F, Ê), a stochastic process B t (ω), t 0 satisfies For each t 1 t n, B tn B tn 1 is independent to (B t1,, B tn 1 ). B t is identically distributed as B s+t B s, for all s, t 0

78 Theorem If, under some (Ω, F, Ê), a stochastic process B t (ω), t 0 satisfies For each t 1 t n, B tn B tn 1 is independent to (B t1,, B tn 1 ). B t is identically distributed as B s+t B s, for all s, t 0 Ê[ B t 3 ] = o(t).

79 Theorem If, under some (Ω, F, Ê), a stochastic process B t (ω), t 0 satisfies For each t 1 t n, B tn B tn 1 is independent to (B t1,, B tn 1 ). B t is identically distributed as B s+t B s, for all s, t 0 Ê[ B t 3 ] = o(t). Then B is a G-Brownian motion.

80 G Brownian Fact Like N (0, [σ 2, σ 2 ])-distribution, the G-Brownian motion B t (ω) =ω t, t 0, can strongly correlated under the unknown objective probability, it can even be have very long memory. But it is i.i.d under the robust expectation Ê. By which we can have many advantages in analysis, calculus and computation, compare with, e.g. fractal B.M.

81 Itô s integral of G Brownian motion For each process (η t ) t 0 L 2,0 F (0, T ) of the form η t (ω) = N 1 j=0 ξ j (ω)i [tj,t j+1 )(t), ξ j L 2 (F tj ) (F tj -meas. & Ê[ ξ j 2 ] < ) we define I (η) = T 0 η(s)db s := N 1 j=0 ξ j (B tj+1 B tj ).

82 Itô s integral of G Brownian motion For each process (η t ) t 0 L 2,0 F (0, T ) of the form η t (ω) = N 1 j=0 ξ j (ω)i [tj,t j+1 )(t), ξ j L 2 (F tj ) (F tj -meas. & Ê[ ξ j 2 ] < ) we define I (η) = T 0 η(s)db s := N 1 j=0 ξ j (B tj+1 B tj ). Lemma We have and T Ê[( 0 T Ê[ 0 η(s)db s ) 2 ] η(s)db s ]=0 T 0 Ê[(η(t)) 2 ]dt.

83 Definition Under the Banach norm η 2 := T 0 Ê[(η(t)) 2 ]dt, I (η) : L 2,0 (0, T ) L 2 (F T ) is a contract mapping We then extend I (η) to L 2 (0, T ) and define, the stochastic integral T 0 η(s)db s := I (η), η L 2 (0, T ).

84 Lemma We have (i) t s η udb u = r s η udb u + t r η udb u. (ii) t s (αη u + θ u )db u = α t s η udb u + t s θ udb u, α L 1 (F s ) (iii) Ê[X + T r η u db u H s ]=Ê[X ], X L 1 (F s ).

85 Quadratic variation process of G BM We denote: B t = B 2 t 2 t 0 B s db s = N 1 lim max(t k+1 t k ) 0 k=0 (B t N k+1 B tk ) 2 B is an increasing process called quadratic variation process of B. Ê[ B t ]=σ 2 t but Ê[ B t ]= σ 2 t

86 Quadratic variation process of G BM We denote: B t = B 2 t 2 t 0 B s db s = N 1 lim max(t k+1 t k ) 0 k=0 (B t N k+1 B tk ) 2 B is an increasing process called quadratic variation process of B. Ê[ B t ]=σ 2 t but Ê[ B t ]= σ 2 t Lemma B s t := B t+s B s, t 0 is still a G-Brownian motion. We also have B t+s B s B s t U([σ 2 t, σ 2 t]).

87 We have the following isometry T Ê[( 0 T η(s)db s ) 2 ]=Ê[ η 2 (s)d B s ], 0 η MG 2 (0, T )

88 Itô s formula for G Brownian motion X t = X 0 + t 0 α s ds + t 0 η s d B s + t 0 β s db s

89 Itô s formula for G Brownian motion X t = X 0 + t 0 α s ds + t 0 η s d B s + t 0 β s db s Theorem. Let α, β and η be process in L 2 G (0, T ). Then for each t 0 and in L 2 G (H t) we have Φ(X t )=Φ(X 0 )+ + t 0 t 0 Φ x (X u )β u db u + t 0 Φ x (X u )α u du [Φ x (X u )η u Φ xx(x u )β 2 u]d B u

90 Stochastic differential equations Problem We consider the following SDE: X t = X 0 + t 0 b(x s )ds + t 0 h(x s )d B s + t 0 σ(x s )db s, t > 0. where X 0 R n is given b, h, σ : R n R n are given Lip. functions. The solution: a process X M 2 G (0, T ; Rn ) satisfying the above SDE.

91 Stochastic differential equations Problem We consider the following SDE: X t = X 0 + t 0 b(x s )ds + t 0 h(x s )d B s + t 0 σ(x s )db s, t > 0. where X 0 R n is given b, h, σ : R n R n are given Lip. functions. The solution: a process X M 2 G (0, T ; Rn ) satisfying the above SDE. Theorem There exists a unique solution X M 2 G (0, T ; Rn ) of the stochastic differential equation.

92 Prospectives Risk measures and pricing under dynamic volatility uncertainties ([A-L-P1995], [Lyons1995]) for path dependent options; Stochastic (trajectory) analysis of sublinear and/or nonlinear Markov process.

93 Prospectives Risk measures and pricing under dynamic volatility uncertainties ([A-L-P1995], [Lyons1995]) for path dependent options; Stochastic (trajectory) analysis of sublinear and/or nonlinear Markov process. New Feynman-Kac formula for fully nonlinear PDE: path-interpretation. t u(t, x) =Ê x [(B t ) exp( 0 c(b s )ds)] t u = G (D 2 u)+c(x)u, u t=0 = ϕ(x).

94 Prospectives Risk measures and pricing under dynamic volatility uncertainties ([A-L-P1995], [Lyons1995]) for path dependent options; Stochastic (trajectory) analysis of sublinear and/or nonlinear Markov process. New Feynman-Kac formula for fully nonlinear PDE: path-interpretation. t u(t, x) =Ê x [(B t ) exp( 0 c(b s )ds)] t u = G (D 2 u)+c(x)u, u t=0 = ϕ(x). Fully nonlinear Monte-Carlo simulation.

95 Prospectives Risk measures and pricing under dynamic volatility uncertainties ([A-L-P1995], [Lyons1995]) for path dependent options; Stochastic (trajectory) analysis of sublinear and/or nonlinear Markov process. New Feynman-Kac formula for fully nonlinear PDE: path-interpretation. t u(t, x) =Ê x [(B t ) exp( 0 c(b s )ds)] t u = G (D 2 u)+c(x)u, u t=0 = ϕ(x). Fully nonlinear Monte-Carlo simulation. BSDE driven by G-Brownian motion: a challenge.

96 The talk is based on: Peng, S. (2006) G Expectation, G Brownian Motion and Related Stochastic Calculus of Ito s type, in arxiv:math.pr/ v2 3Jan 2006, to appear in Proceedings of 2005 Abel Symbosium, Springer.

97 The talk is based on: Peng, S. (2006) G Expectation, G Brownian Motion and Related Stochastic Calculus of Ito s type, in arxiv:math.pr/ v2 3Jan 2006, to appear in Proceedings of 2005 Abel Symbosium, Springer. Peng, S. (2006) Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation, in arxiv:math.pr/ v2 28Jan 2006.

98 The talk is based on: Peng, S. (2006) G Expectation, G Brownian Motion and Related Stochastic Calculus of Ito s type, in arxiv:math.pr/ v2 3Jan 2006, to appear in Proceedings of 2005 Abel Symbosium, Springer. Peng, S. (2006) Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation, in arxiv:math.pr/ v2 28Jan Peng, S. Law of large numbers and central limit theorem under nonlinear expectations, in arxiv:math.pr/ v1 13 Feb 2007 Peng, S. A New Central Limit Theorem under Sublinear Expectations, arxiv: v1 [math.pr] 18 Mar 2008

99 The talk is based on: Peng, S. (2006) G Expectation, G Brownian Motion and Related Stochastic Calculus of Ito s type, in arxiv:math.pr/ v2 3Jan 2006, to appear in Proceedings of 2005 Abel Symbosium, Springer. Peng, S. (2006) Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation, in arxiv:math.pr/ v2 28Jan Peng, S. Law of large numbers and central limit theorem under nonlinear expectations, in arxiv:math.pr/ v1 13 Feb 2007 Peng, S. A New Central Limit Theorem under Sublinear Expectations, arxiv: v1 [math.pr] 18 Mar 2008 Peng, S.L. Denis, M. Hu and S. Peng, Function spaces and capacity related to a Sublinear Expectation: application to G-Brownian Motion Pathes, see arxiv: v1 [math.pr] 9 Feb 2008.

100 The talk is based on: Peng, S. (2006) G Expectation, G Brownian Motion and Related Stochastic Calculus of Ito s type, in arxiv:math.pr/ v2 3Jan 2006, to appear in Proceedings of 2005 Abel Symbosium, Springer. Peng, S. (2006) Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation, in arxiv:math.pr/ v2 28Jan Peng, S. Law of large numbers and central limit theorem under nonlinear expectations, in arxiv:math.pr/ v1 13 Feb 2007 Peng, S. A New Central Limit Theorem under Sublinear Expectations, arxiv: v1 [math.pr] 18 Mar 2008 Peng, S.L. Denis, M. Hu and S. Peng, Function spaces and capacity related to a Sublinear Expectation: application to G-Brownian Motion Pathes, see arxiv: v1 [math.pr] 9 Feb Song, Y. (2007) A general central limit theorem under Peng s G-normal distribution, Preprint.

101 Thank you,

102 In the classic period of Newton s mechanics, including A. Einstein, people believe that everything can be deterministically calculated. The last century s research affirmatively claimed the probabilistic behavior of our universe: God does plays dice! Nowadays people believe that everything has its own probability distribution. But a deep research of human behaviors shows that for everything involved human or life such, as finance, this may not be true: a person or a community may prepare many different p.d. for your selection. She change them, also purposely or randomly, time by time.